Tsu-ku (つく, 突く), tsu-ku (つく, 着く), tsu-ku (つく, 付く)

つく(tsu-ku) has versatile meanings. By using Chinese characters we can find seemingly different meanings.

突く (つく, tsu-ku) - to push with some tool with a point. Transitive verb.
着く (つく, tsu-ku) - to arrive. Intransitive verb.
付く (つく, tsu-ku) - to attach. Intransitive verb. The transitive verb form is 付ける (つける, tsu-ke-ru).
就く (つく, tsu-ku) - to kake (a post). Intransitive verb.
衝く (つく, tsu-ku) - to attack at a narrow area. The transitive verb.
尽く (つく, tsu-ku) - to use up, to reach at the end. Note: now usually 尽きる (つく, tsu-ki-ru) is used. Both 尽く (つく, tsu-ku) and 尽きる (つく, tsu-ki-ru)  are intransitive verbs. The transitive verb form is 尽くす (つくす, tsu-ku-su).
憑く (つく, tsu-ku) - to attach, mentally or psychologically. Intransitive verb.
搗く (つく, tsu-ku) - to pound (dough, rice, etc). Transitive verb.
点く (つく, tsu-ku) - to light up, to catch fire. 着く or 付く is also used for this use. Intransitive verb.

There are several ways to sort them out into two groups in term of meanings. One way is to divide them, in terms of grammar, into 1) Transitive verb and 2) Intransitive verb.

1) Transitive verbs - taking を（wo, o) showing a direct object
突く (つく, tsu-ku) - to push with some tool with a point
衝く (つく, tsu-ku) - to attack at a narrow area
搗く (つく, tsu-ku) - to pound (dough, rice)

The general meaning of these is <to hit at a point>.

2) Intransitive verbs - taking に (ni) showing an indirect object, usually place

着く (つく, tsu-ku) - to arrive at
付く (つく, tsu-ku) - to attach (to be attached), to stick to. Intransitive verb in Japanese.
就く (つく, tsu-ku) - to kake (a post) - Intransitive verb in Japanese.<to take a seat> ＜席につく(seki-ni-tsuku)＞, <to take a sea of king> ＜王位につく(oui-ni-tsuku)＞.
尽く (つく, tsu-ku) - to use up (to be used up), to reach to the limit
憑く (つく, tsu-ku) - to attach at (to), to be attached, mentally or psychologically. Intransitive verb in Japanese.
点く (つく, tsu-ku) - to light up, to catch fire

The general meaning of these is< to come(get) to a point>. The last one つく(tsu-ku) <to light up, to catch fire> may be regarded as <to happen (or some change) at a point>.


sptt

Shiru-shi (しるし), kata-chi (かたち), kimo-chi (きもち)

When Japanese people give a present to someone they used to say "This is just a sign (しるし, shirushi) for appreciation)". "for appreciation" is often omitted.. しるし(shi-ru-shi) can be replaced by きもち(気持ち, kimochi; mind, not feeling in this case). But these expressions have been less and less used as modern Japanese people think this expression not straightforward to express thanks, which may have been mostly influenced by the American gift-giving culture. The actual gift is a material so they tried to avoid to refer to material things, which meant mean for them.

Apart from the above gift-giving cultural custom, let's think about the above native Japanese word しるし (shirushi). which is used also against material things. きもち(気持ち, kimochi, "mind") is against material things too. There is another word かたち(katachi) - form, which is used also against material things but used in some other expressions.

しるし (shirushi) also means "mark" besides "sign". The verb form is しるす (shiru-su) is "to mark". The related word しらせ(shirase) means "news" or  "notice" - besides "form" it. Its verb form しらせる(shirase-ru) means "to make something known". しらす(shira-su) is also used but less common now.

かたち(katachi) means "figure", "shape" besides "form". The word may have derived from かた (kata), which also means "form" or "mold(mould). かたい (kata-i) is an adjective form meaning "hard". かたち(katachi) and かた(kata) generally refer to a solid form, figure, shape. The verb form of  かたい (kata-i) is かためる (kata-meru) meaning  "to solidify". There seems no simple verb form of かたち(katachi) (to form, to shape) we use かたちづくる(katachi-zukuru), literally "form - make".


sptt